Third-Order Noncanonical Neutral Delay Differential Equations: Nonexistence of Kneser Solutions via Myshkis Type Criteria
Abstract
:1. Introduction
- and
- and f does not vanish identically;
- , and is in noncanonical form, that is,
- such that and are commute.
2. Main Results
3. Examples
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
- Dzurina, J.; Jadlovska, I. Oscillation of third-order differential equations with noncanonical operators. Appl. Math. Comput. 2018, 336, 394–402. [Google Scholar] [CrossRef]
- Kiguradze, I.; Chauturia, T. Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1993. [Google Scholar]
- Trench, W.F. Canonical forms and principal systems for general disconjugate equations. Trans. Am. Math. Soc. 1974, 184, 319–327. [Google Scholar] [CrossRef]
- Baculikova, B. Asymptotic properties of noncanonical third order differential equations. Math. Slovaca 2019, 6, 1341–1350. [Google Scholar] [CrossRef]
- Alrashdi, H.S.; Moaaz, O.; Alqawasm, K.; Kanan, M.; Zakarya, M.; Elabbasy, E.M. Asymptotic and oscillatory properties of third-order differential equations with multiple delays in noncanonical case. Mathematics 2024, 12, 1189. [Google Scholar] [CrossRef]
- Alzabut, J.; Grace, S.R.; Santra, S.S.; Chhatria, G.N. Asymptotic and oscillatory behavior of third-order nonlinear differential equations with canonical operators and mixed neutral terms. Qual. Theory Dyn. Syst. 2023, 22, 15. [Google Scholar] [CrossRef]
- Baculikova, B.; Dzurina, J. Oscillation of third-order neutral differential equations. Math. Comput. Model. 2010, 52, 215–226. [Google Scholar] [CrossRef]
- Baculikova, B.; Dzurina, J. Remarks on properties of Kneser solutions for third-order neutral differential equations. Appl. Math. Lett. 2017, 63, 1–5. [Google Scholar] [CrossRef]
- Baculikova, B.; Rani, B.; Selvarangam, S.; Thandapani, E. Properties of Kneser’s solution for half-linear third-order neutral differential equations. Acta Math. Hungar. 2017, 152, 525–533. [Google Scholar] [CrossRef]
- Chatzarakis, G.E.; Dzurina, J.; Jadlovska, I. Oscillatory properties of third-order neutral delay differential equations with noncanonical operators. Mathematics 2019, 7, 1177. [Google Scholar] [CrossRef]
- Dosla, Z.; Liska, P. Oscillation of third-order nonlinear neutral differential equations. Appl. Math. Lett. 2016, 56, 42–48. [Google Scholar] [CrossRef]
- Dzurina, J.; Thandapani, E.; Tamilvanan, S. Oscillation of solutions to third-order half-linear neutral differential equations. Electron. J. Differ. Equ. 2012, 2012, 1–9. [Google Scholar]
- Dzurina, J.; Grace, S.R.; Jadlovska, I. On nonexistence of Kneser solutions of third-order neutral delay differential equations. Appl. Math. Lett. 2019, 88, 193–200. [Google Scholar] [CrossRef]
- Feng, L.; Han, Z. Oscillation of a class of third-order neutral differential equations with noncanonical opertors. Bull. Malays. Math. Sci. Soc. 2021, 44, 2519–2530. [Google Scholar] [CrossRef]
- Graef, J.R.; Tunc, E.; Grace, S.R. Oscillatory and asymptotic brhavior of a third-order nonlinear neutral differential equation. Opusc. Math. 2017, 37, 839–852. [Google Scholar] [CrossRef]
- Hassan, T.S.; El-Matary, B.M. Asymptotic behavior and oscillation of third-order nonlinear neutral differential equations with mixed nonlinearities. Mathematics 2023, 11, 424. [Google Scholar] [CrossRef]
- Jadlovska, I.; Chatzarakis, G.E.; Dzurina, J.; Grace, S.R. On sharp oscillation criteria for general third-order delay differential equations. Mathematics 2021, 9, 1675. [Google Scholar] [CrossRef]
- Kitamura, Y.; Kusano, T. Oscillation of first order nonlinear differential equations with deviating arguments. Proc. Am. Math. Soc. 1980, 78, 61–68. [Google Scholar]
- Li, T.; Zhang, C.; Xing, G. Oscillation of third-order neutral delay differential equations. Abst. Appl. Anal. 2012, 2012, 569201. [Google Scholar] [CrossRef]
- Li, T.; Rogovchenko, Y.V. On the asymptotic behvior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Math. Lett. 2020, 105, 106293. [Google Scholar] [CrossRef]
- Philos, C.G. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. 1981, 36, 168–178. [Google Scholar] [CrossRef]
- Thandapani, E.; Li, T. On the osillation of third-order quasilinear neutral functional differential equations. Arch. Math. 2011, 47, 181–199. [Google Scholar]
- Myshkis, A.D. Linear Differential Equations with Retarded Argument; Izdat. Nauka: Moscow, Russia, 1972. [Google Scholar]
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Nithyakala, G.; Chatzarakis, G.E.; Ayyappan, G.; Thandapani, E. Third-Order Noncanonical Neutral Delay Differential Equations: Nonexistence of Kneser Solutions via Myshkis Type Criteria. Mathematics 2024, 12, 2847. https://doi.org/10.3390/math12182847
Nithyakala G, Chatzarakis GE, Ayyappan G, Thandapani E. Third-Order Noncanonical Neutral Delay Differential Equations: Nonexistence of Kneser Solutions via Myshkis Type Criteria. Mathematics. 2024; 12(18):2847. https://doi.org/10.3390/math12182847
Chicago/Turabian StyleNithyakala, Gunasekaran, George E. Chatzarakis, Govindasamy Ayyappan, and Ethiraju Thandapani. 2024. "Third-Order Noncanonical Neutral Delay Differential Equations: Nonexistence of Kneser Solutions via Myshkis Type Criteria" Mathematics 12, no. 18: 2847. https://doi.org/10.3390/math12182847
APA StyleNithyakala, G., Chatzarakis, G. E., Ayyappan, G., & Thandapani, E. (2024). Third-Order Noncanonical Neutral Delay Differential Equations: Nonexistence of Kneser Solutions via Myshkis Type Criteria. Mathematics, 12(18), 2847. https://doi.org/10.3390/math12182847